A physics (projectile motion) question that I'm sure someone could figure out in two seconds.

[Random Variable]

I'm stumped.

An incline makes an angle of "Ø" with the positive x-axis. A ball is thrown upwards at the base of the incline at an angle of "ß" with respect to the positive x-axis. (In other words, the angle between the incline and the velocity vector of the ball is ß- Ø.) If the ball is to land up the incline as far as possible (radially), why should angle "ß" be equal to (1/2)*arctan[-cotØ]?

EDIT: It actually takes about 20 minutes to solve the problem.

 

[Yossarian]

This is a job for Captain Derivative!

 

[RaynorWolfcastle]

reminds me of a question I had in mechanics last semester where you had a boy standing on a hill of angle A. He throws a ball with initial velocity V. What is the angle theta that maximizes the distance the ball travels?

I solved it by interpretting the path of the ball as a parabola and the hill as a line. Then after a couple of pages of calculus and trig you come up with something vaguley similar to what you wrote. I think I may still have the maple file somehwere on my hard disk

 

[LSUfan]

The answer is always c if you dont know it

 

[RaynorWolfcastle]

Originally posted by: LSUfan
The answer is always c if you dont know it

even when it's not a multiple choice question?

 

[slikmunks]

i could figure it out, but it would require getting a pencil and paper to help you figure it out... and i'm feeling sooooo lazy right now... sorry dude... next time

 

[josphII]

what do you mean up the incline?? you mean the furthest distance radially from the origin, or the max distance H, from the X axis?

 

[VBboy]

I'd say that the angle B should be 45 degrees regardless of the other angle. 45 degrees is what gives you the largest landing distance...

 

[josphII]

the answer doesnt even make sense, the angles is negative for 0 - 90 degrees

 

[Random Variable]

Originally posted by: RaynorWolfcastle
reminds me of a question I had in mechanics last semester where you had a boy standing on a hill of angle A. He throws a ball with initial velocity V. What is the angle theta that maximizes the distance the ball travels?

I solved it by interpretting the path of the ball as a parabola and the hill as a line. Then after a couple of pages of calculus and trig you come up with something vaguley similar to what you wrote. I think I may still have the maple file somehwere on my hard disk

The problem shouldn't require much calculus (if any).

what do you mean up the incline?? you mean the furthest distance radially from the origin, or the max distance H, from the X axis?

Radially.

I'd say that the angle B should be 45 degrees regardless of the other angle. 45 degrees is what gives you the largest landing distance...

But if the angle of the incline is 60°, for example, you would have just thrown the ball directly into the incline.

 

[Random Variable]

I thought the answer would simply be [90-Ø]/2.

 

[Random Variable]

Originally posted by: josphII
the answer doesnt even make sense, the angles is negative for 0 - 90 degrees

Yeah, that doesn't make sense. The angle would be in the fourth quadrant.

 

[Random Variable]

I solved it by interpretting the path of the ball as a parabola and the hill as a line. Then after a couple of pages of calculus and trig you come up with something vaguley similar to what you wrote. I think I may still have the maple file somehwere on my hard disk

You made the incline the x-axis?

 

[silverpig]

Draw it out on some graph paper. Your incline will be a line intersecting the origin, with equation y = -kx where k = slope = y/x = arctan(theta)

Your projectile's path will be y = -c(x-a)^2 + b or, parametrically,

y(t) = v(sin(B))*t - 1/2*-9.81*t^2
x(t) = v(cos(B)*t)

You must find B such that the intersection of the parabola and the line is a maximum.

Just a few hints to get you started...

 

[RaynorWolfcastle]

Heh, I found the solution on the course website (I did it differently, complicating my life significantly, I remember now)

Solution

It is almost exactly the same as your question but for a ball being thrown down a hill instead of up a hill

edit (I also found the Maple file I used to solve the problem. It's pretty messy but I set up a plot that you can configure with different angles to show you where the ball would land)

 

[MichaelD]

How do you make those cool letters in your post? The fancy "A" and "B" are the shiznitz.

 

[RaynorWolfcastle]

Originally posted by: MichaelD
How do you make those cool letters in your post? The fancy "A" and "B" are the shiznitz.

that's an alpha and a beta, greek letters. I think there are ASCII codes for them

 

[Random Variable]

Originally posted by: RaynorWolfcastle

Originally posted by: MichaelD
How do you make those cool letters in your post? The fancy "A" and "B" are the shiznitz.

that's an alpha and a beta, greek letters. I think there are ASCII codes for them

Actually, it's phi and beta.

 

[RaynorWolfcastle]

Originally posted by: Vespasian

Originally posted by: RaynorWolfcastle

Originally posted by: MichaelD
How do you make those cool letters in your post? The fancy "A" and "B" are the shiznitz.

that's an alpha and a beta, greek letters. I think there are ASCII codes for them

Actually, it's phi and beta.

Sorry about that, I didn't reread the post, I just assumed that A = alpha and B = beta

 

[Random Variable]

Originally posted by: RaynorWolfcastle
Heh, I found the solution on the course website (I did it differently, complicating my life significantly, I remember now)

Solution

It is almost exactly the same as your question but for a ball being thrown down a hill instead of up a hill

edit (I also found the Maple file I used to solve the problem. It's pretty messy but I set up a plot that you can configure with different angles to show you where the ball would land)

Thank you. I'll see if it helps.